Problem 16 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

USING EQUATIONS Which of the following are asymptotes of the graph of \(y=3 \tan 4 x\) ? (A) \(x=\frac{\pi}{8}\) (B) \(x=\frac{\pi}{4}\) (C) \(x=0\) (D) \(x=-\frac{5 \pi}{8}\)

Step-by-Step Solution

Answer

The correct asymptotes for the function \(y=3 \tan 4x\) are \(x= \frac{\pi}{8}\) and \(x= -\frac{5\pi}{8}\). Options A and D are correct.

Step 1: Find the General Form Asymptote

First, determine the general form of the vertical asymptotes. For the general function \(y = a \tan(bx)\), the vertical asymptotes occur at \(x = \frac{(2n+1) \pi}{2b}\) where \(n\) is an integer.

Step 2: Apply this to Our Function

To find the asymptotes for the function \(y=3 \tan(4x)\), replace \(b\) with 4 in the general form equation. So, the vertical asymptotes occur at \(x = \frac{(2n + 1) \pi}{8}\). The denominator is 8 because \(2b = 2*4 = 8\).

Step 3: Compare with Option Asymptotes

Now, we examine the given options to see if they fit our formula. \n\n(A) \(x=\frac{\pi}{8}\) : This fits the formula when n = 0 and so this is an asymptote. \n\n(B) \(x= \frac{\pi}{4}\) : This doesn't fit the formula and so is not an asymptote. \n\n(C) \(x=0\) : This doesn't fit the formula and so is not an asymptote. \n\n(D) \(x= -\frac{5\pi}{8}\) : This fits the formula when n = -3, and so this is an asymptote.

Key Concepts

Trigonometric Functions

Trigonometric functions are a group of functions often used in mathematics to study angles, triangles, and periodic phenomena. These functions include sine, cosine, tangent, and their respective reciprocals: cosecant, secant, and cotangent. They are fundamental in both pure and applied mathematics. One core characteristic of trigonometric functions is their periodic nature, which means they repeat their values in regular intervals. The most common interval for repetition corresponds to the unit circle, where angles are measured in radians. For instance, sine and cosine functions have a period of \(2\pi\), whereas the tangent and cotangent functions have a period of \(\pi\).

Key features of trigonometric functions include:

• Amplitude (the height of the peak).
• Period (the distance over which the function repeats).
• Phase shift (the horizontal shift from the origin).
• Vertical shift (the movement of the function upwards or downwards).

Together, these elements help define the shape and position of the trigonometric graphs, allowing us to understand and predict changes in terms of angles and distances.

Vertical Asymptotes

In mathematics, an asymptote is a line that a graph approaches but never touches. A vertical asymptote is a vertical line where the function goes towards infinity or negative infinity, creating a dramatic spike in the graph.

Vertical asymptotes occur in rational functions, trigonometric functions, and other types of functions typically where the denominator of the function equals zero. Analyzing the behavior of the graph near these lines helps us understand how the function behaves as it approaches specific values of \(x\).

For the tangent function often denoted as \(y = \tan(bx)\), vertical asymptotes occur at points where the function is undefined. This typically happens because the tan function involves dividing by cosine, which equals zero at these points. In the general form, "\(y = a \tan(bx)\), vertical asymptotes occur at \(x = \frac{(2n+1)\pi}{2b}\), where \(n\) is an integer. This formula helps identify the specific locations of the vertical asymptotes on the graph.

Tangent Function

The tangent function, one of the primary trigonometric functions, is essential for understanding ratios of angles. Defined as the ratio of sine over cosine, \(\tan(x) = \frac{\sin(x)}{\cos(x)}\), it behaves differently from sine and cosine, which range between -1 and 1. The tangent function has an infinite range and includes both positive and negative values.

The graph of the tangent function exhibits a pattern of repeating waves, which demonstrate certain characteristics:

• Period of \(\pi\): Unlike sine and cosine, which have a \(2\pi\) period, tangent repeats every \(\pi\).
• Vertical asymptotes: Occurring at points where \(\cos(x) = 0\). These correspond to odd multiples of \(\frac{\pi}{2}\) in the basic \(\tan(x)\) function. When transformed, the positions shift according to modifications made by scale factors or horizontal shifts.
• No amplitude: Because the function doesn't have a maximum or minimum height, we don't describe it using amplitude in the same manner as sine or cosine.

Knowing these properties helps in sketching and interpreting the graph of the tangent function, especially when dealing with transformations such as \(y = 3\tan(4x)\), which involves both vertical stretching and horizontal compression.